Introduction to Dynamic Macroeconomic General Equilibrium Models

By José Luis Torres Chacon

This book offers an introductory step-by-step course in Dynamic Stochastic General Equilibrium (DSGE) modelling. Modern macroeconomic analysis is increasingly concerned with the construction, calibration and/or estimation and simulation of DSGE models. The book is intended for graduate students as an introductory course to DSGE modelling and for

• Language: English
• Publisher: Vernon Press
• Release date: May 13, 2016
• ISBN: 9781622730452

José Luis Torres Chacon

José L. Torres is Associate Professor of Economics, Head of the Department of Economics, Faculty of Economics, University of Málaga (Spain). His current research areas include: Technological Change, Economic Growth, Dynamic General Equilibrium modelling. He has published several books and a large number of papers in journals as Information Economics and Policy, Public Choice, Macroeconomic Dynamics, Journal of Macroeconomics, Eastern European Economics, Empirical Economics, Economic Modelling, Open Economies Review, Economic Letters, Journal of International Financial Markets, Institutions and Money, SERIES, and Economic Issues.

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Introduction

The Dynamic Stochastic General Equilibrium (DSGE) approach is a cornerstone of modern macroeconomic analysis. The development of DSGE models in macroeconomics is mainly a response to two long-lasting challenges: The need to address the Lucas critique (Lucas, 1976) and the desire to build micro-founded macroeconomic models. Three main terms define this theoretical approach: Dynamic (D), Stochastic (S), and General Equilibrium (GE).

One of the objectives of economic analysis is to understand how an economy works and to carry out experiments to study the effects of a particular change or disturbance on the economy. This kind of analysis presents enormous difficulties due to the complexity of the phenomena we want to explain. In other fields such as physics or chemistry, which in some ways can be thought as analogous to economics, researchers have experimental laboratories in which to replicate the conditions that exist in the real world and thus, perform experiments using scale models. In fact, this has long been the goal of economic analysis; to build scale models of the real world with which to perform a number of experiments and to know in advance the effects of certain disturbances or changes in economic policies at the macroeconomic level.¹

However, when we compare economics to either physics or chemistry we should keep in mind that although the analytical tools can be similar, there is an important difference: the human factor. In the economy, the effects of a particular disturbance will be determined by the decisions taken by economic agents and how they react to the disturbance. This difference between the economy and other experimental sciences is a major obstacle to the construction of macroeconomic laboratories in which to perform scale experiments that can subsequently be transferred to the real world.

Currently, macroeconomic analysis has a widely accepted theoretical scheme, the DSGE framework, which avails a scale model economy and, therefore, a laboratory in which to carry out experiments. This theoretical framework, initially developed by Ramsey in the late 1920s (Ramsey, 1927, 1928), has been widely adopted today as one of the main tools for macroeconomic modelling. DSGE models represent a scale model of the real world that can be considered as too aggregated and too simple, but very useful to study how the economy responds to different shocks or to quantify the effects of monetary and fiscal policy. The success of neoclassical DSGE models as the basic framework for macroeconomic analysis is based on the following three main characteristics that render that theoretical setting a valid representation of reality.

First, the outcome of the model depends on the decisions taken by the economic agents. Instead of modelling markets, as has been done traditionally, the general equilibrium neoclassical model focuses on the behavior of three main types of economic agents: households, firms and the government, but it may include additional economic agents such as a central bank or the foreign sector. The basic idea is to determine the basic rules of behavior of the different economic agents and then observe how they react to changes in the environment. The equilibrium results from the combination of economic decisions taken by all economic agents.

Second, it is a general equilibrium model. For a macroeconomic model to be a valid replica of an economy, such a model must consider the multiple and complex relationships between the different economic variables. In reality, all macroeconomic variables are related to each other, either directly or indirectly, so we must abandon that pipe dream of ceteris paribus, which does nothing but hinder our understanding of how the economy works.

Finally, it is a dynamic model in which time plays a fundamental role. This ingredient is very important, because when a disturbance hits the economy, macroeconomic variables do not return to the equilibrium instantaneously, but they change very slowly over time, producing complex reactions in the economy as a whole. Furthermore, in our model economy we need to consider investment decisions, which are of great importance for the economy but only makes sense in a dynamic context.

1.1

Macroeconomic DSGE Modelling

The general strategy used by current applied macroeconomics for both disturbance and policy analysis and forecasting, is the construction of formal structures through equations that reflect the interrelationships between the different economic variables. These simplified structures is what we call a model. The essential question here is not that these theoretical constructions are realistic descriptions of the economy, but that they are able to explain the dynamics observed in the economy.

After a long period characterized by a deep rift between macroeconomics and microeconomics, current developments in macroeconomics are based on the microeconomic analysis of economic agents decisions. It is not intended that macroeconomics goes down to the decisions of individual consumers or firms, but it is important that macroeconomic theoretical frameworks be consistent with the underlying behavior of millions of consumers and millions of firms that inhabit an economy, and in this sense we speak about the microfoundations of modern macroeconomics. These microfoundations have created a rigorous formal theoretical setting that we call modern macroeconomics, the workhorse of which is the neoclassical growth model based on Ramsey.

Modern macroeconomics uses formal and rigorous models in order to provide explanations of different problems observed in real economies, with the aim to offer solutions or policy recommendations to prevent these problems or alleviate their consequences on social welfare. Current macroeconomics is formalized through mathematical models and subject to the traditional scientific method of measurement, theory and validation. Measurement, which is a description of the facts, is a necessary step of any economic analysis, but the description of an observed phenomenon does not itself constitute an explanation of it. For that a second step is necessary: the development of a theory. Although data can be tortured using a large variety of statistical and econometric techniques, they will not confess. Data will only speak through theoretical models. The third step is the hardest. The theoretical models are based on abstract assumptions which represent a simplification of reality, but the important thing is that they can be useful to offer a valid explanation of an economic fact. Therefore, it is not possible to reject a model ex ante because it is based on assumptions that we believe not too realistic. Rather, the validation must be based on the usefulness of these models to explain reality, and whether they are more useful than other models (Canova, 2007).

Our macroeconomic laboratory consists of a DSGE model (and a computer with appropriate software). This model economy is a necessary simplification of reality. The reason we trust them is easy to understand. Think of a street plan. A street plan is a simplification of a city, but it is an extremely useful tool to move in it. A street plan includes a number of non-real assumptions: The scale is different to the real one, is a two-dimensional representation of a three-dimensional space, is totally flat, etc. The lack of realism of a street plan does not hinder its effectiveness. What makes street plans useful is the proper matching of symbolic elements on the plan with the actual layout of the elements to which the plan refers. In the same way, the degree of realism offered by an economic model is not a goal to be pursued by macroeconomists, but rather the model’s usefulness in explaining macroeconomic reality.

Overall, the structure of a macroeconomic model is similar to those models used to explain the behavior of a physical system, except for one important difference: the behavior of the economy depends on decisions taken by humans. In a physical system, the particles are neutral with respect to the laws driving their dynamics and interactions. In the economy, particles (economic agents) have theories about how the system they belong to works, and they make decisions that affect the dynamics of the system.

The basic structure of a macroeconomic model can be defined in terms of the following system of equations:

where Xt is a vector of endogenous variables, Zt a vector of exogenous variables, Et is the expectation operator, and ut is a vector of random disturbances with proper density functions. The function F(⋅) is what we call economic theory. The solution to this system of stochastic equations would be a sequence of probability distributions. This system of equations contains a key element: the value of the endogenous variables in a given period of time depends on its future expected value.

The use of theoretical models to describe and understand the behavior of an economy is important for a variety of reasons:

First, theoretical models are important to understand the complex relationships between macroeconomic variables which cannot be observed just by looking directly at the data. Data only speaks through models.

Theoretical models introduce a metric to talk about the economy in commonly understandable terms and to define non-observable variables, such as the marginal productivity of capital, or state variables, such as total factor productivity.

Theoretical models can be used to make simulations for policy analysis and counterfactual experiments.

Finally, forecasting is only possible by using a theoretical model (structural forecasting approach).

As discussed above, macroeconomic analysis depends on the availability of a laboratory in which an artificial economy can be simulated in an attempt to replicate certain phenomena we observe in reality. This artificial economy is based on the construction of a theoretical macroeconomic model. The main theoretical framework we use in current macroeconomic analysis is the neoclassical dynamic general equilibrium growth model. The basis of this model is not new, as it was developed by Ramsey in the late 1920s. This model is easy to understand: it is an economy in which there are three (although they may be other economic agents) types of economic agents: households, firms, and the government (only the first two in the simplest version). Households take decisions in terms of how much to consume (save) and how much time is devoted to work (leisure). Firms decide how much they will produce. Equilibrium of the economy will be defined by a situation in which all decisions taken by all economic agents are compatible and feasible.

With this theoretical framework at hand it is possible to obtain numerical solutions for the steady state and for the dynamics of the variables by calibrating or estimating the model for a given economy. National Accounts will provide the necessary information to calibrate or estimate the parameters of the model. Therefore, there must be a direct correspondence between National Accounts and the DSGE model. If this is the case, we already have our macroeconomic laboratory.

1.2

DSGE software

DSGE modelling requires the use of numerical solution methods. This means that once a particular DSGE model has been built up, to make it quantitatively operative, we need software and hardware. Taking theoretical models to the computer is a compulsory step for DSGE modelling. Whereas in the past this was a very difficult task, currently we can find a large number of publicly available software tools for DSGE modelling written in different computer languages, such as Matlab, R, Gauss, Mathematica, C, etc. Most of these tools can be found in DSGE-NET, which is an International Network for DSGE modelling, monetary and fiscal policy,² or in the QM&RBC (Quantitative Macroeconomics and Real Business Cycle) page by Christian Zimmermann.³ General software platforms for DSGE modelling are, for instance, Dynare, gEcon and IRIS.

Dynare is a pre-processor that uses a very simple language that allows the conversion of a DSGE model in a program that can be implemented in various programming languages (Matlab or Octave) to solve, estimate and simulate the model.⁴ The source syntax is very friendly and simple. We only need to provide the set of endogenous variables, the set of exogenous variables, the parameters, the value of the parameters and the equations of the model. This software platform can use data to estimate the parameters of DSGE model, using both maximum likelihood or Bayesian techniques. Dynare has been developed at CEPREMAP, by a team directed by Michel Julliard, Stéphane Adjemian and Sébastien Villemot. This is the tool used in this book.

Another software tool for solving DSGE models is gEcon.⁵ This tool has been developed in R by the Department for Strategic Analyses at the Chancellery of the Prime Minister of the Republic of Poland by Grzegorz Klima, Karol Podemski and Kaja Retkiewicz-Wijtiwiak. The main characteristic of gEcon is that the model can be solved directly by writing the optimization problems for the economic agents. That is, it is not necessary to derive first order conditions and equilibrium equations. gEcon implements an algorithm for automatic derivation of first order conditions, steady state and linearization matrices.

Finally, IRIS is a toolbox in Matlab for macroeconomic modelling and forecasting, developed by the IRIS Solutions Team since 2001, headed by Jaromír Beneš.⁶IRIS can solve, simulate, and estimate (using maximum likelihood methods) a DSGE model. Forecasting using the structural model is also allowed.

1.3

Book organization

All chapters in this book follow a similar structure. In each chapter a particular DSGE model is developed, introducing a relevant topic in the basic DSGE model. Equilibrium conditions are obtained and parameters calibrated. Then, we study the effects of a shock and compute impulse-response functions of the macroeconomic variables. This exercise is done using Dynare for Matlab. Each chapter includes an appendix with the corresponding Dynare code.

Chapter 2 presents the basic dynamic general equilibrium model, considering the behavior of two economic agents: Households and firms. Here we show the basic structure of the model used in current macroeconomics. The structure of this model is very simple (although even simpler versions are possible). Households make decisions about how much to consume (how much to save) and how many hours they will devote to work (or to leisure), given the price of the production factors, in order to maximize lifetime utility. Firms decide how much labor and capital will be hired to maximize profits. Once equilibrium of the model economy is obtained and the parameters calibrated, this framework can be used to perform a variety of simulation exercises. In our setting, the simulation exercise will study how the economy responds to an aggregate productivity shock, that is, the prototype RBC analysis.

Chapter 3 introduces consumption habit formation as an extension to the basic DSGE model developed in the previous chapter. In the standard neoclassical DSGE model, utility function is instantaneous time-separable. This means that current utility only depends on current consumption and does not depend on the level of consumption in previous periods. However, empirical evidence shows the existence of habit formation which implies that utility function is not time-separable. This can explain one observed deviation from the permanent income-life cycle hypothesis: the excess smoothness of consumption with respect to changes in income.

Chapter 4 develops a DSGE model in which a portion of the population cannot make optimal decisions regarding their consumption path because they cannot borrow, that is, they cannot bring future income to the current period. This is the case when there are liquidity constraints and imperfect financial markets. The purpose of introducing liquidity constraints is to explain another observed deviation from the permanent income-life cycle hypothesis: The excess sensitivity of consumption to current income. The model assumes that the economy is composed of two types of agents: Ricardian agents, who have no liquidity constraints and can take optimal decisions regarding consumption-saving path, and non-Ricardian agents, who are subject to liquidity constraints, and consumption of each period is restricted by their income in that period. The aggregate behavior of the economy is given by the weighted sum of the behavior of each group of agents.

Chapter 5 takes into account the existence of adjustment costs in the investment process, taking as reference the Tobin’s Q theory. Variations in the physical capital stock of the economy are subject to adjustment costs that could be important in explaining investment decisions. These costs may be associated to the existing capital stock and/or to investment. The DSGE model developed in this chapter considers the existence of adjustment costs associated with investment. The simulation exercise will show how investment decisions react to a Total Factor Productivity shock when investment adjustment costs are present.

Chapter 6 studies the role of investment-specific technological progress. Standard DSGE model considers a single source of technological progress: Total Factor Productivity changes or neutral technological progress. However, physical capital assets are not homogeneous over time and new vintages of capital assets are an important source of technological progress. New capital assets incorporate an improved technology compared to previously existing assets. This type of technological progress is associated to the investment process. The model includes two types of technological change: total factor productivity (TFP) or neutral technological change and investment-specific technological (ISTC) change.

Chapter 7 introduces a new economic agent in the basic DSGE model: The government. The government can be introduced in the standard DSGE model is a large variety of ways. This chapter considers the role of the government as a tax-levying entity. In particular, the model incorporates three types of taxes: consumption tax, labor income tax, and capital income tax. For simplicity, it is assumed that the government returns fiscal revenues as lump-sum transfers. A number of exercises are conducted using this theoretical framework: computation of Laffer curves, changes in taxes, and a productivity shock.

Chapter 8 continues with the role of the public sector, but incorporating to the previous model the existence of public spending. In this model the households’ utility depends not only on their consumption of private goods, but also depends on the consumption of goods provided